Constraining the $\overline{K}N$ interaction from the $1S$ level shift of kaonic deuterium

Motivated by the precise measurement of the $1S$ level shift of kaonic hydrogen, we perform accurate three-body calculations for the spectrum of kaonic deuterium using a realistic antikaon-nucleon $\left(\overline{K}N\right)$ interaction. In order to describe both short- and long-range behaviors of the kaonic atomic states, we solve the three-body Schrödinger equation with a superposition of a large number of correlated Gaussian basis functions covering distances up to several hundreds of fm. Transition energies between $1S,2P$, and $2S$ states are determined with high precision. The complex energy shift of the $1S$ level of kaonic deuterium is found to be $\mathrm{\Delta }E-i\mathrm{\Gamma }/2=(670-i508)$ eV. The sensitivity of this level shift with respect to the isospin $I=1$ component of the $\overline{K}N$ interaction is examined. It is pointed out that an experimental determination of the kaonic deuterium level shift within an uncertainty of 25% will provide a constraint for the $I=1$ component of the $\overline{K}N$ interaction significantly stronger than that from kaonic hydrogen.